Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M \neq N2 and M2 = N4, then

Engineering

Mathematics

Inverse of a Matrix

Question

Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M $\neq$ N² and M² = N⁴, then

determinant of (M² + MN²) $\geq$ 1.

determinant of (M² + MN²) is 0.

there is a 3 × 3 non‑zero matrix U such that (M² + MN²) U is the zero matrix.

for a 3 × 3 matrix U, if (M² + MN²) U equals the zero matrix then U is the zero matrix.

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M² – N⁴ = O ⇒ (M – N²) (M + N²) = O ……(1)

$∴$ |M – N²| |M + N²| = 0

So, |M + N²| = 0 ⇒ |M| |M + N²| = 0 ⇒ |M² + MN²| = 0 ⇒ Option (A) is correct

(If |M + N²| $\neq$ 0 then (M + N²)^–1 exist. So, from eq (1), M – N² = O, which is contradiction)

Also from Eq.(1), (M + N²) (M – N²) = O

⇒ (M² + MN²) (M – N²) = O ⇒ (M² + MN²) U = O, where U = M – N² ⇒ Option (B) is correct